Small Number Of Replicates Research Articles

How many replicate tests are needed to test cookstove performance and emissions? — Three is not always adequate

Almost half of the world's population still cooks on biomass cookstoves of poor efficiency and primitive design, such as three stone fires (TSF). Emissions from biomass cookstoves contribute to adverse health effects and climate change. A number of improved cookstoves with higher energy efficiency and lower emissions have been designed and promoted across the world. During the design development, and for the selection of a stove for dissemination, the stove performance and emissions are commonly evaluated, communicated and compared using the arithmetic average of replicate tests made using a standardized laboratory-based test, commonly the water boiling test (WBT). However, the statistics section of the test protocol contains some debatable concepts and in certain cases, easily misinterpreted recommendations. Also, there is no agreement in the literature on how many replicate tests should be performed to ensure “confidence” in the reported average performance (with three being the most common number of replicates). This matter has not received sufficient attention in the rapidly growing literature on stoves, and yet is crucial for estimating and communicating the performance of a stove, and for comparing the performance between stoves. We illustrate an application using data from a number of replicate tests of performance and emission of the Berkeley–Darfur Stove (BDS) and the TSF under well-controlled laboratory conditions. Here we focus on two as illustrative: time-to-boil and emissions of PM2.5 (particulate matter less than or equal to 2.5μm in diameter). We demonstrate that an interpretation of the results comparing these stoves could be misleading if only a small number of replicates had been conducted. We then describe a practical approach, useful to both stove testers and designers, to assess the number of replicates needed to obtain useful data from previously untested stoves with unknown variability.

How many replicate tests are needed to test cookstove performance and emissions? — Three is not adequate

Almost half of the world's population still cooks on biomass cookstoves of poor efficiency and primitive design, such as three-stone fire (TSF). Emissions from biomass cookstoves contribute to adverse health effects and climate change. A number of improved cookstoves with higher energy efficiency and lower emissions have been designed and promoted across the world. During the design development, and for the selection of a stove for dissemination, the stove performance and emissions are commonly evaluated, communicated and compared using the arithmetic average of replicate tests made using a standardized laboratory-based test, commonly the water boiling test (WBT). However, published literature shows different WBT results reported from different laboratories for the same stove technology. Also, there is no agreement in the literature on how many replicate tests should be performed to ensure “confidence” in the reported average performance (with three being the most common number of replicates). This matter has not received sufficient attention in the rapidly growing literature on stoves, and yet is crucial for estimating and communicating the performance of a stove, and for comparing the performance between stoves. We illustrate an application using data from a number of replicate tests of performance and emission of the Berkeley–Darfur Stove (BDS) and the TSF under well-controlled laboratory conditions. Here we focus on two as illustrative: time-to-boil and emissions of PM2.5 (particulate matter less than or equal to 2.5μm in diameter). We demonstrate that interpretation of the results comparing these stoves could be misleading if only a small number of replicates had been conducted. We then describe a practical approach, useful to both stove testers and designers, to assess the number of replicates needed to obtain useful data from previously untested stoves with unknown variability.

A NEW METHOD FOR NONHOMOGENEOUS TIME COURSE EXPRESSION ANALYSIS

Time course expression analysis constitutes a large portion of applications of microarray experiments. One primary goal of such experiments is to detect genes with the temporal changes over a period of time or at some interested time points. Difficulties arising from data with small number of replicates over only a few unaligned time points in multiple groups pose challenges for efficient statistical analysis. Some known methods are limited by the unverifiable assumptions or by the scope of applications for only two groups. We present a new method for detecting differentially expressed genes under nonhomogeneous time course experiments in multiple groups. The new method first models the time course curve of one gene by a Gaussian process to align the nonhomogeneous time course data and to compute the gradient of the time course curve as well, the latter of which is used as directional information to enhance the sensitivity of detection for temporal changes. Second, we adopt a nonparametric method to test a surrogate hypothesis based on the augmented data from the Gaussian process model. The proposed method is robust in terms of model fitting and testing. It does not require any distributional assumption for the observations or the test statistic and the method works for the case with as few as triplicate samples over four or five time points under multiple groups. We show the effectiveness and superiority of the new method in comparison with some existing methods using simulated models and two real data sets.

Hybrid method for the analysis of time series gene expression data

Time series analysis plays an increasingly important role in the study of gene expression data. Some problems, such as a large amount of noise and a small number of replicates, are computational challenges in time series expression data analysis. This paper proposes a hybrid method for analyzing time series gene expression data (HMTS). In the HMTS method, we employ a combination of K-means clustering, regression analysis and piecewise polynomial curve fitting. The K-means clustering procedure is used to divide noisy time series into different clusters, and regression analysis is used to delete outliers according to different clusters. All time series data are divided into multiple segmentations, and polynomial curve fitting is used to fit all segmentation data. The HMTS method can obtain good estimates, especially when there is noise in the data.

Elements to Improve Galling Resistance Test Results Using the ASTM G98 Method

Abstract The ASTM G98 Standard Test Method for Galling Resistance of Materials describes a relatively easy and quick method to screen materials for galling resistance. It has been used for several decades by material producers to report the galling resistance of their various grades of material. Design engineers subsequently use the data to make informed decisions regarding material selection. One of the most attractive features of the method is the small number of replicates that need to be run to estimate galling resistance. In the minimal case, only two replicates need to be tested. In recent years, however, research has been conducted that exposed several issues with the ASTM G98 method that can lead to reduced accuracy in results. These issues have included stress concentrations in the contact region, inconsistent oxidation layer formation on specimens, and small sampling sets leading to statistically insignificant results, among other issues. In this paper, several problematic issues are described and discussed. Suggestions are given to improve the repeatability of test results while still conforming to the standardized test method.

Optimal significance analysis of microarray data in a class of tests whose null statistic can be constructed

Microarray data often consist of a large number of genes and a small number of replicates. We have examined testing the null hypothesis of equality of mean for detecting differentially expressed genes. The p-value for each gene is often estimated using permutation samples not only for the target gene but also for other genes. This method has been widely used and discussed. However, direct use of the permutation method for the p-value estimation may not work well, because two types of genes are mixed in the sample; some genes are differentially expressed, whereas others are not. To overcome this difficulty, various methods for appropriately generating null permutation samples have been proposed. In this paper, we consider two classes of test statistics that are naturally modified to null statistics. We then obtain the uniformly most powerful (UMP) unbiased tests among these classes. If the underlying distribution is symmetric, the UMP unbiased test statistic is similar to that proposed by Pan (Bioinformatics 19:1333–1340, 2003). Under another condition, the UMP unbiased test statistic has a different formula with one more degree of freedom and therefore is expected to give a more powerful test and a more accurate p-value estimation from a modified null statistic. In microarray data, because the number of replicates is often small, differences in the degree of freedom will produce large effects on the power of test and the variance of the p-value estimation. Some simulation studies and real data analyses are illustrated to investigate the performances of the methods.

Comparison of classical statistical, bayes, and empirical bayes methods in analysis of microarray data; which method can deal with large number of variables (genes) and small number of replications?

Comparison of classical statistical, bayes, and empirical bayes methods in analysis of microarray data; which method can deal with large number of variables (genes) and small number of replications?

Double error shrinkage method for identifying protein binding sites observed by tiling arrays with limited replication

ChIP-chip has been widely used for various genome-wide biological investigations. Given the small number of replicates (typically two to three) per biological sample, methods of analysis that control the variance are desirable but in short supply. We propose a double error shrinkage (DES) method by using moving average statistics based on local-pooled error estimates which effectively control both heterogeneous error variances and correlation structures of an extremely large number of individual probes on tiling arrays. Applying DES to ChIP-chip tiling array study for discovering genome-wide protein-binding sites, we identified 8400 target regions that include highly likely TFIID binding sites. About 33% of these were well matched with the known transcription starting sites on the DBTSS library, while many other newly identified sites have a high chance to be real binding sites based on a high positive predictive value of DES. We also showed the superior performance of DES compared with other commonly used methods for detecting actual protein binding sites.

Multiple life stages with multiple replicated density levels are required to estimate density dependence for plants

Negative density dependence resulting from intraspecific competition can regulate plant populations by limiting demographic rates (survival, growth, fecundity). However, the strength of intraspecific competition can vary within and among populations due to spatial or temporal environmental heterogeneity, or genetic differences. Quantification of variation under a relatively constant environment is needed to assess the inherent potential for density dependence to vary. This knowledge will help adjust data collection effort required for parameterisation of density dependence. Our review of published plant demographic studies revealed that only half of the studies included the whole life‐cycle in the analysis of density dependence. Approximately half of the studies manipulated density, while the rest examined density dependence from observed densities in the field. Regardless of the design used, density dependence was estimated from a small number of replicates. To investigate inherent variation in density dependence during the life‐cycle, and the effect of low replication on density dependence estimates, we combined an experimental approach with simulations for an invasive herb Senecio madagascariensis. We found significant negative density dependence for five out of six examined demographic rates in a constant environment, with the strength of density dependence increasing during the life‐cycle. An exception was plant growth, in which the direction of density dependence varied from positive to negative depending on the life stage. Simulations showed substantial deviation for density dependence parameterised from a small number of replicates even when environmental variation was minimal. This suggests that data collection procedures currently used to assess the effect of density on plant demographic rates may produce inaccurate estimates, increasing uncertainty in demographic models. Due to variation in the direction and strength of density dependence during the life‐cycle, multiple life stages with multiple replicated density levels are required to parameterise density dependence for demographic rates.

Partition clustering of high dimensional low sample size data based on [formula omitted]-values

Clustering techniques play an important role in analyzing high dimensional data that is common in high-throughput screening such as microarray and mass spectrometry data. Effective use of the high dimensionality and some replications can help to increase clustering accuracy and stability. In this article a new partitioning algorithm with a robust distance measure is introduced to cluster variables in high dimensional low sample size (HDLSS) data that contain a large number of independent variables with a small number of replications per variable. The proposed clustering algorithm, PPCLUST, considers data from a mixture distribution and uses p -values from nonparametric rank tests of homogeneous distribution as a measure of similarity to separate the mixture components. PPCLUST is able to efficiently cluster a large number of variables in the presence of very few replications. Inherited from the robustness of rank procedure, the new algorithm is robust to outliers and invariant to monotone transformations of data. Numerical studies and an application to microarray gene expression data for colorectal cancer study are discussed.

Optimising sample volume and replicates using the Bou-Rouch method for the rapid assessment of hyporheic fauna

Despite the widespread use of the Bou-Rouch method in obtaining hyporheic samples in ecological studies, problems persist in comparing data because of differences in volume and number of samples taken in various studies. Towards standardising this method, we conducted a study in the Delour River, Ireland, between March and September 2006. The hyporheic habitat was divided into three lateral zones: the flowing stream (SS); the stream–terrestrial ecotone (EC); and the terrestrial margin (TM). Between 3 and 6 random insertions of the stand pipe were made into the hyporheic habitat at 0.2 m and 0.5 m in each zone. Ten consecutive 1-L samples were pumped at each point to determine an optimum sample volume and replicates needed to estimate taxon richness and abundance. The optimum sample combinations range between 6 and 15 3- to 10-L samples depending on depth and habitat zone. Our results show that both spatial and temporal scales are important factors in considering the optimum combinations of sample volume and number of independent spatial replicates needed to sample stream hyporheos, with the zone closer to the flowing stream requiring many small-volume samples whereas areas further away towards the alluvial groundwater need larger volumes with small numbers of replicates.

A preliminary analysis of mate choice in a bush cricket (Poecilimon laevissimus: Tettigoniidae) suggests virginity is more important than body size

Insects are predicted to prefer larger partners for a number of reasons relating to fitness. In species where males provide an expensive nuptial gift, male and female preferences for a larger partner are likely to be more pronounced. In nuptial-feeding insects however, models of sperm competition and female choice predict that males and females should also prefer virgin partners. Here we test the relative importance of size vs virginity in a Greek bushcricket, Poecilimon laevissimus, in which males offer typically large nuptial gifts during mating. While only a small number of replicates could be implemented, and there is a clear need for further analysis, we found that all males and females preferred to mate with virgins, despite the fact that nearly 90% of the virgins were smaller in size than the nonvirgins offered. In terms of mate choice, virginity therefore appears more important than body size in P. laevissimus.

Empirical Bayes analysis of unreplicated microarray data

Because of the high costs of microarray experiments and the availability of only limited biological materials, microarray experiments are often performed with a small number of replicates. Investigators, therefore, often have to perform their experiments with low replication or without replication. However, the heterogeneous error variability observed in microarray experiments increases the difficulty in analyzing microarray data without replication. No current analysis techniques are practically applicable to such microarray data analysis. We here introduce a statistical method, the so-called unreplicated heterogeneous error model (UHEM) for the microarray data analysis without replication. This method is possible by utilizing many adjacent-intensity genes for estimating local error variance after nonparametric elimination of differentially expressed genes between different biological conditions. We compared the performance of UHEM with three empirical Bayes prior specification methods: between-condition local pooled error, pseudo standard error, or adaptive standard error-based HEM. We found that our unreplicated HEM method is effective for the microarray data analysis when replication of an array experiment is impractical or prohibited.

The “Window t test”: a simple and powerful approach to detect differentially expressed genes in microarray datasets

Abstract This work focuses on differential expression analysis of microarray datasets. One way to improve such statistical analyses is to integrate biological information in the design of these analyses. In this paper, we will use the relationship between the level of gene expression and variability. Using this biological information, we propose to integrate the information from multiple genes to get a better estimate of individual gene variance, when a small number of replicates are available, to increase the power of the statistical analysis. We describe a strategy named the “Window t test” that uses multiple genes which share a similar expression level to compute the variance which is then incorporated a classic t test. The performances of this new method are evaluated by comparison with classic and widely-used methods for differential expression analysis (the classic Student t test, the Regularized t test (reg t test), SAM, Limma, LPE and Shrinkage t). In each case tested, the results obtained were at least equivalent to the best performing method and, in most cases, outperformed it. Moreover, the Window t test relies on a very simple procedure requiring small computing power compared with other methods designed for microarray differential expression analysis.

Statistical identification of differentially labeled peptides from liquid chromatography tandem mass spectrometry

LC-MS/MS with certain labeling techniques such as isotope-coded affinity tag (ICAT) enables quantitative analysis of paired protein samples. However, current identification and quantification of differentially expressed peptides (and proteins) are not reliable for large proteomics screening of complex biological samples. The number of replicates is often limited because of the high cost of experiments and the limited supply of samples. Traditionally, a simple fold change cutoff is used, which results in a high rate of false positives. Standard statistical methods such as the two-sample t-test are unreliable and severely underpowered due to high variability in LC-MS/MS data, especially when only a small number of replicates are available. Using an advanced error pooling technique, we propose a novel statistical method that can reliably identify differentially expressed proteins while maintaining a high sensitivity, particularly with a small number of replicates. The proposed method was applied both to an extensive simulation study and a proteomics comparison between microparticles (MPs) generated from platelet (platelet MPs) and MPs isolated from plasma (plasma MPs). In these studies, we show a significant improvement of our statistical analysis in the identification of proteins that are differentially expressed but not detected by other statistical methods. In particular, several important proteins - two peptides for beta-globin and three peptides for von Willebrand Factor (vWF) - were identified with very small false discovery rates (FDRs) by our method, while none was significant when other conventional methods were used. These proteins have been reported with their important roles in microparticles in human blood cells: vWF is a platelet and endothelial cell product that binds to P-selectin, GP1b, and GP IIb/IIIa, and beta-globin is one of the peptides of hemoglobin involved in the transportation of oxygen by red blood cells.

A meta-data based method for DNA microarray imputation

BackgroundDNA microarray experiments are conducted in logical sets, such as time course profiling after a treatment is applied to the samples, or comparisons of the samples under two or more conditions. Due to cost and design constraints of spotted cDNA microarray experiments, each logical set commonly includes only a small number of replicates per condition. Despite the vast improvement of the microarray technology in recent years, missing values are prevalent. Intuitively, imputation of missing values is best done using many replicates within the same logical set. In practice, there are few replicates and thus reliable imputation within logical sets is difficult. However, it is in the case of few replicates that the presence of missing values, and how they are imputed, can have the most profound impact on the outcome of downstream analyses (e.g. significance analysis and clustering). This study explores the feasibility of imputation across logical sets, using the vast amount of publicly available microarray data to improve imputation reliability in the small sample size setting.ResultsWe download all cDNA microarray data of Saccharomyces cerevisiae, Arabidopsis thaliana, and Caenorhabditis elegans from the Stanford Microarray Database. Through cross-validation and simulation, we find that, for all three species, our proposed imputation using data from public databases is far superior to imputation within a logical set, sometimes to an astonishing degree. Furthermore, the imputation root mean square error for significant genes is generally a lot less than that of non-significant ones.ConclusionSince downstream analysis of significant genes, such as clustering and network analysis, can be very sensitive to small perturbations of estimated gene effects, it is highly recommended that researchers apply reliable data imputation prior to further analysis. Our method can also be applied to cDNA microarray experiments from other species, provided good reference data are available.

Optimal Shrinkage Estimation of Variances With Applications to Microarray Data Analysis

Microarray technology allows a scientist to study genomewide patterns of gene expression. Thousands of individual genes are measured with a relatively small number of replications, which poses challenges to traditional statistical methods. In particular, the gene-specific estimators of variances are not reliable and gene-by-gene tests have low powers. In this article we propose a family of shrinkage estimators for variances raised to a fixed power. We derive optimal shrinkage parameters under both Stein and squared loss functions. Our results show that the standard sample variance is inadmissible under either loss function. We propose several estimators for the optimal shrinkage parameters and investigate their asymptotic properties under two scenarios: large number of replications and large number of genes. We conduct simulations to evaluate the finite sample performance of the data-driven optimal shrinkage estimators and compare them with some existing methods. We construct F-like statistics using these shrinkage variance estimators and apply them to detect differentially expressed genes in a microarray experiment. We also conduct simulations to evaluate performance of these F-like statistics and compare them with some existing methods.

Use of Optical Density Detection Times To Assess the Effect of Acetic Acid on Single-Cell Kinetics

The growth of Listeria innocua at different acetic acid concentrations (0 to 2,000 ppm) was monitored by optical density measurements in a Bioscreen (Labsystems, Vantaa, Finland). The generated populations came from low inocula that were obtained by serial dilution. A new method to estimate both the growth rate and the lag time of single cells from the detection times (time to reach an optical density of 0.11) was developed. It assumes that the single-cell lag times follow a gamma distribution and takes into account the randomness of the inoculation level. (The initial cell number per well was assumed to follow a Poisson distribution.) In this way, relatively small numbers of replicates are sufficient to obtain a robust estimation of the distribution of single-cell lag times. The results were validated with plate count experiments. It was found that logarithms of both the growth rates and of population lag times increased linearly with the acetic acid concentration. The logarithm of the scale parameter of the gamma distribution of the single-cell lag times also increased linearly with the acetic acid concentration irrespective of the phase of the inoculum.

Replicated batch means for steady-state simulations

This paper studies a new steady-state simulation output analysis method called replicated batch means in which a small number of replications are conducted and the observations in these replications are grouped into batches. This paper also introduces and compares methods for selecting the initial state of each replication. More specifically, we show that confidence intervals constructed by the replicated batch means method are valid for large batch sizes and derive expressions for the expected values and variances of the steady-state mean and variance estimators for stationary processes and large sample sizes. We then use these expressions, analytical examples, and numerical experiments to compare the replicated batch means method with the standard batch means and multiple replications methods. The numerical results, which are obtained from an AR(1) process and a small, nearly-decomposable Markov chain, show that the multiple replications method often gives confidence intervals with poorer coverage than the standard and replicated batch means methods and that the replicated batch means method, implemented with good choices of initialization method and number of replications, provides confidence interval coverages that range from being comparable with to being noticeably better than coverages obtained by the standard batch means method. © 2006 Wiley Periodicals, Inc. Naval Research Logistics, 2006

Statistically Consistent Identification of Differentially Expressed Genes in DNA Chip Data Over the Whole Expression Range

It is a well known problem that standard techniques for analysing DNA chip data misspecify genes. In particular, genes that are confirmed to be active, often do not show up as potential candidates. This is possibly due to non-homogeneous distributions of expression levels over the whole expression range. We introduce a method that allows the detection of genes based on a self-adaptive threshold. The threshold is determined for equally-populated expression bands by assuming a normal distribution of logarithms of expression level ratios. By specifying a significance level, the threshold is set according to 'local' expression statistics within a band. We call this method the relative variance method (RVM). We derive a test statistic for the RVM and compare it with other methods. On this statistical basis, we show that RVM is a complementary approach to the t-test, significance analysis of microarrays (SAM) or empirical Bayes analysis of microarrays (EBAM). The RVM should be particularly useful for experiments with small sample size. Using a clinical dataset, we demonstrate that the RVM can correctly identify known marker genes, which are not found by the t-test, SAM or EBAM. In situations with limited sample material and small number of replicates, as is often the case in clinical datasets, use of the proposed RVM provides a higher reliability of potential candidate genes.